Properties

Label 1800.1.ba.b
Level $1800$
Weight $1$
Character orbit 1800.ba
Analytic conductor $0.898$
Analytic rank $0$
Dimension $4$
Projective image $D_{3}$
CM discriminant -8
Inner twists $8$

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Newspace parameters

Level: \( N \) \(=\) \( 1800 = 2^{3} \cdot 3^{2} \cdot 5^{2} \)
Weight: \( k \) \(=\) \( 1 \)
Character orbit: \([\chi]\) \(=\) 1800.ba (of order \(6\), degree \(2\), not minimal)

Newform invariants

Self dual: no
Analytic conductor: \(0.898317022739\)
Analytic rank: \(0\)
Dimension: \(4\)
Relative dimension: \(2\) over \(\Q(\zeta_{6})\)
Coefficient field: \(\Q(\zeta_{12})\)
Defining polynomial: \(x^{4} - x^{2} + 1\)
Coefficient ring: \(\Z[a_1, a_2]\)
Coefficient ring index: \( 1 \)
Twist minimal: no (minimal twist has level 72)
Projective image: \(D_{3}\)
Projective field: Galois closure of 3.1.648.1
Artin image: $C_{12}\times S_3$
Artin field: Galois closure of \(\mathbb{Q}[x]/(x^{24} - \cdots)\)

$q$-expansion

The \(q\)-expansion and trace form are shown below.

\(f(q)\) \(=\) \( q -\zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{4} q^{6} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} +O(q^{10})\) \( q -\zeta_{12}^{5} q^{2} + \zeta_{12}^{5} q^{3} -\zeta_{12}^{4} q^{4} + \zeta_{12}^{4} q^{6} -\zeta_{12}^{3} q^{8} -\zeta_{12}^{4} q^{9} + \zeta_{12}^{2} q^{11} + \zeta_{12}^{3} q^{12} -\zeta_{12}^{2} q^{16} -\zeta_{12}^{3} q^{17} -\zeta_{12}^{3} q^{18} + q^{19} + \zeta_{12} q^{22} + \zeta_{12}^{2} q^{24} + \zeta_{12}^{3} q^{27} -\zeta_{12} q^{32} -\zeta_{12} q^{33} -\zeta_{12}^{2} q^{34} -\zeta_{12}^{2} q^{36} -\zeta_{12}^{5} q^{38} -\zeta_{12}^{4} q^{41} -\zeta_{12}^{5} q^{43} + q^{44} + \zeta_{12} q^{48} -\zeta_{12}^{4} q^{49} + \zeta_{12}^{2} q^{51} + \zeta_{12}^{2} q^{54} + \zeta_{12}^{5} q^{57} + \zeta_{12}^{4} q^{59} - q^{64} - q^{66} + \zeta_{12} q^{67} -\zeta_{12} q^{68} -\zeta_{12} q^{72} + \zeta_{12}^{3} q^{73} -\zeta_{12}^{4} q^{76} -\zeta_{12}^{2} q^{81} -\zeta_{12}^{3} q^{82} + 2 \zeta_{12}^{5} q^{83} -\zeta_{12}^{4} q^{86} -\zeta_{12}^{5} q^{88} -2 q^{89} + q^{96} + \zeta_{12}^{5} q^{97} -\zeta_{12}^{3} q^{98} + q^{99} +O(q^{100})\)
\(\operatorname{Tr}(f)(q)\) \(=\) \( 4q + 2q^{4} - 2q^{6} + 2q^{9} + O(q^{10}) \) \( 4q + 2q^{4} - 2q^{6} + 2q^{9} + 2q^{11} - 2q^{16} + 4q^{19} + 2q^{24} - 2q^{34} - 2q^{36} + 2q^{41} + 4q^{44} + 2q^{49} + 2q^{51} + 2q^{54} - 2q^{59} - 4q^{64} - 4q^{66} + 2q^{76} - 2q^{81} + 2q^{86} - 8q^{89} + 4q^{96} + 4q^{99} + O(q^{100}) \)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/1800\mathbb{Z}\right)^\times\).

\(n\) \(577\) \(901\) \(1001\) \(1351\)
\(\chi(n)\) \(-1\) \(-1\) \(\zeta_{12}^{4}\) \(-1\)

Embeddings

For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.

For more information on an embedded modular form you can click on its label.

Label \(\iota_m(\nu)\) \( a_{2} \) \( a_{3} \) \( a_{4} \) \( a_{5} \) \( a_{6} \) \( a_{7} \) \( a_{8} \) \( a_{9} \) \( a_{10} \)
499.1
−0.866025 0.500000i
0.866025 + 0.500000i
−0.866025 + 0.500000i
0.866025 0.500000i
−0.866025 + 0.500000i 0.866025 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 0
499.2 0.866025 0.500000i −0.866025 + 0.500000i 0.500000 0.866025i 0 −0.500000 + 0.866025i 0 1.00000i 0.500000 0.866025i 0
1699.1 −0.866025 0.500000i 0.866025 + 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
1699.2 0.866025 + 0.500000i −0.866025 0.500000i 0.500000 + 0.866025i 0 −0.500000 0.866025i 0 1.00000i 0.500000 + 0.866025i 0
\(n\): e.g. 2-40 or 990-1000
Significant digits:
Format:

Inner twists

Char Parity Ord Mult Type
1.a even 1 1 trivial
8.d odd 2 1 CM by \(\Q(\sqrt{-2}) \)
5.b even 2 1 inner
9.c even 3 1 inner
40.e odd 2 1 inner
45.j even 6 1 inner
72.p odd 6 1 inner
360.z odd 6 1 inner

Twists

       By twisting character orbit
Char Parity Ord Mult Type Twist Min Dim
1.a even 1 1 trivial 1800.1.ba.b 4
5.b even 2 1 inner 1800.1.ba.b 4
5.c odd 4 1 72.1.p.a 2
5.c odd 4 1 1800.1.bk.d 2
8.d odd 2 1 CM 1800.1.ba.b 4
9.c even 3 1 inner 1800.1.ba.b 4
15.e even 4 1 216.1.p.a 2
20.e even 4 1 288.1.t.a 2
35.f even 4 1 3528.1.cg.a 2
35.k even 12 1 3528.1.ba.a 2
35.k even 12 1 3528.1.ce.b 2
35.l odd 12 1 3528.1.ba.b 2
35.l odd 12 1 3528.1.ce.a 2
40.e odd 2 1 inner 1800.1.ba.b 4
40.i odd 4 1 288.1.t.a 2
40.k even 4 1 72.1.p.a 2
40.k even 4 1 1800.1.bk.d 2
45.j even 6 1 inner 1800.1.ba.b 4
45.k odd 12 1 72.1.p.a 2
45.k odd 12 1 648.1.b.b 1
45.k odd 12 1 1800.1.bk.d 2
45.l even 12 1 216.1.p.a 2
45.l even 12 1 648.1.b.a 1
60.l odd 4 1 864.1.t.a 2
72.p odd 6 1 inner 1800.1.ba.b 4
80.i odd 4 1 2304.1.o.c 4
80.j even 4 1 2304.1.o.c 4
80.s even 4 1 2304.1.o.c 4
80.t odd 4 1 2304.1.o.c 4
120.q odd 4 1 216.1.p.a 2
120.w even 4 1 864.1.t.a 2
180.v odd 12 1 864.1.t.a 2
180.v odd 12 1 2592.1.b.a 1
180.x even 12 1 288.1.t.a 2
180.x even 12 1 2592.1.b.b 1
280.y odd 4 1 3528.1.cg.a 2
280.bp odd 12 1 3528.1.ba.a 2
280.bp odd 12 1 3528.1.ce.b 2
280.br even 12 1 3528.1.ba.b 2
280.br even 12 1 3528.1.ce.a 2
315.bs even 12 1 3528.1.ba.a 2
315.bt odd 12 1 3528.1.ba.b 2
315.cb even 12 1 3528.1.cg.a 2
315.cg even 12 1 3528.1.ce.b 2
315.ch odd 12 1 3528.1.ce.a 2
360.z odd 6 1 inner 1800.1.ba.b 4
360.bo even 12 1 72.1.p.a 2
360.bo even 12 1 648.1.b.b 1
360.bo even 12 1 1800.1.bk.d 2
360.br even 12 1 864.1.t.a 2
360.br even 12 1 2592.1.b.a 1
360.bt odd 12 1 216.1.p.a 2
360.bt odd 12 1 648.1.b.a 1
360.bu odd 12 1 288.1.t.a 2
360.bu odd 12 1 2592.1.b.b 1
720.cn odd 12 1 2304.1.o.c 4
720.cp even 12 1 2304.1.o.c 4
720.cr odd 12 1 2304.1.o.c 4
720.ct even 12 1 2304.1.o.c 4
2520.hj odd 12 1 3528.1.ba.a 2
2520.hq even 12 1 3528.1.ba.b 2
2520.hw odd 12 1 3528.1.cg.a 2
2520.iy odd 12 1 3528.1.ce.b 2
2520.jd even 12 1 3528.1.ce.a 2
    
        By twisted newform orbit
Twist Min Dim Char Parity Ord Mult Type
72.1.p.a 2 5.c odd 4 1
72.1.p.a 2 40.k even 4 1
72.1.p.a 2 45.k odd 12 1
72.1.p.a 2 360.bo even 12 1
216.1.p.a 2 15.e even 4 1
216.1.p.a 2 45.l even 12 1
216.1.p.a 2 120.q odd 4 1
216.1.p.a 2 360.bt odd 12 1
288.1.t.a 2 20.e even 4 1
288.1.t.a 2 40.i odd 4 1
288.1.t.a 2 180.x even 12 1
288.1.t.a 2 360.bu odd 12 1
648.1.b.a 1 45.l even 12 1
648.1.b.a 1 360.bt odd 12 1
648.1.b.b 1 45.k odd 12 1
648.1.b.b 1 360.bo even 12 1
864.1.t.a 2 60.l odd 4 1
864.1.t.a 2 120.w even 4 1
864.1.t.a 2 180.v odd 12 1
864.1.t.a 2 360.br even 12 1
1800.1.ba.b 4 1.a even 1 1 trivial
1800.1.ba.b 4 5.b even 2 1 inner
1800.1.ba.b 4 8.d odd 2 1 CM
1800.1.ba.b 4 9.c even 3 1 inner
1800.1.ba.b 4 40.e odd 2 1 inner
1800.1.ba.b 4 45.j even 6 1 inner
1800.1.ba.b 4 72.p odd 6 1 inner
1800.1.ba.b 4 360.z odd 6 1 inner
1800.1.bk.d 2 5.c odd 4 1
1800.1.bk.d 2 40.k even 4 1
1800.1.bk.d 2 45.k odd 12 1
1800.1.bk.d 2 360.bo even 12 1
2304.1.o.c 4 80.i odd 4 1
2304.1.o.c 4 80.j even 4 1
2304.1.o.c 4 80.s even 4 1
2304.1.o.c 4 80.t odd 4 1
2304.1.o.c 4 720.cn odd 12 1
2304.1.o.c 4 720.cp even 12 1
2304.1.o.c 4 720.cr odd 12 1
2304.1.o.c 4 720.ct even 12 1
2592.1.b.a 1 180.v odd 12 1
2592.1.b.a 1 360.br even 12 1
2592.1.b.b 1 180.x even 12 1
2592.1.b.b 1 360.bu odd 12 1
3528.1.ba.a 2 35.k even 12 1
3528.1.ba.a 2 280.bp odd 12 1
3528.1.ba.a 2 315.bs even 12 1
3528.1.ba.a 2 2520.hj odd 12 1
3528.1.ba.b 2 35.l odd 12 1
3528.1.ba.b 2 280.br even 12 1
3528.1.ba.b 2 315.bt odd 12 1
3528.1.ba.b 2 2520.hq even 12 1
3528.1.ce.a 2 35.l odd 12 1
3528.1.ce.a 2 280.br even 12 1
3528.1.ce.a 2 315.ch odd 12 1
3528.1.ce.a 2 2520.jd even 12 1
3528.1.ce.b 2 35.k even 12 1
3528.1.ce.b 2 280.bp odd 12 1
3528.1.ce.b 2 315.cg even 12 1
3528.1.ce.b 2 2520.iy odd 12 1
3528.1.cg.a 2 35.f even 4 1
3528.1.cg.a 2 280.y odd 4 1
3528.1.cg.a 2 315.cb even 12 1
3528.1.cg.a 2 2520.hw odd 12 1

Hecke kernels

This newform subspace can be constructed as the intersection of the kernels of the following linear operators acting on \(S_{1}^{\mathrm{new}}(1800, [\chi])\):

\( T_{11}^{2} - T_{11} + 1 \)
\( T_{19} - 1 \)

Hecke characteristic polynomials

$p$ $F_p(T)$
$2$ \( 1 - T^{2} + T^{4} \)
$3$ \( 1 - T^{2} + T^{4} \)
$5$ \( T^{4} \)
$7$ \( T^{4} \)
$11$ \( ( 1 - T + T^{2} )^{2} \)
$13$ \( T^{4} \)
$17$ \( ( 1 + T^{2} )^{2} \)
$19$ \( ( -1 + T )^{4} \)
$23$ \( T^{4} \)
$29$ \( T^{4} \)
$31$ \( T^{4} \)
$37$ \( T^{4} \)
$41$ \( ( 1 - T + T^{2} )^{2} \)
$43$ \( 1 - T^{2} + T^{4} \)
$47$ \( T^{4} \)
$53$ \( T^{4} \)
$59$ \( ( 1 + T + T^{2} )^{2} \)
$61$ \( T^{4} \)
$67$ \( 1 - T^{2} + T^{4} \)
$71$ \( T^{4} \)
$73$ \( ( 1 + T^{2} )^{2} \)
$79$ \( T^{4} \)
$83$ \( 16 - 4 T^{2} + T^{4} \)
$89$ \( ( 2 + T )^{4} \)
$97$ \( 1 - T^{2} + T^{4} \)
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